Elemental and Constraint Equations From a Normal Tree

 

Overview

In order to categorize a system it is important to identify its governing set of equations. While this can be completed according to kinematic/mass acceleration diagrams, it is also possible to formulate these equations according to linear graph theory. This is particularly useful for non-mechanical domains, and for systems. This article assumes a working understanding of across and through variables, and knowledge of how to draw a normal tree.

Example Problem

The electrical system shown below in figure 1 has been processed into a normal tree. Write the system of elemental and constraint equations that describe the system:

Figure 1: Electrical System (left) and corresponding normal tree (right)

First, we will write the set of elemental equations that describe the provided normal tree. There will be one elemental equation per passive element of the tree and each elemental equation will be written in the form of:

$$P=f(S)$$

Where \(P\) (the primary variable) is a function of \(S\) (the secondary variable).

Recall that primary variables are the across variable of twigs and the through variable of links. In the case of our problem we will have the following equations.

$$\begin{array}{l}{I_{R 1}=f\left(V_{R 1}\right)} \\ {I_{R 2}=f\left(V_{R 2}\right)} \\ {V_{C 1}=f\left(I_{C 1}\right)} \\ {V_{C 2}=f\left(l_{C 2}\right)}\end{array}$$

These equations can now be replaced by the corresponding governing equation for each element.

$$\begin{aligned} I_{R 1} &=\left(\frac{1}{R 1}\right) V_{R 1} \\ I_{R 2} &=\left(\frac{1}{R 2}\right) V_{R 2} \\ V_{C 1} &=\left(\frac{1}{C 1}\right) \int I_{C 1} d t \\ V_{C 2} &=\left(\frac{1}{C 2}\right) \int I_{C 2} d t \end{aligned}$$

The equations for \(V_{C 1}\) and \(V_{C 2}\) can also be represented in derivative form if desired.This is often more useful when writing state equations for the system.

While the elemental equations used to describe the system are specific to components but not the system itself, the constraint equations are based in the system's normal graph. There will be one constraint equation for each elemental equation, and each constraint equation will be used to define the secondary variable of an elemental equation.

Hence we will need constraint equations for \(V_{R 1}\), \(V_{R 2}\), \(I_{C 1}\), \(I_{C 2}\).

Process for Writing Constraint (Compatibility) Equations

  1. Cut each passive twig once, cutting only that twig. Do KCL to solve for the secondary variable of that cut twig (through variable).

  2. Remove all links from the normal tree, and temporarily add one link back in at a time. Do KVL on the loop that is formed by the added link to solve for the link's secondary variable (across variable).

Step 1

We first make an imaginary cut through the \(C_2\) twig, and write the corresponding equation.

 
 

This process is repeated for the \(C_1\) twig.

Note that both of these cuts are used to solve for the secondary variable of the cut twig (current).

Step 2

All links are removed and the \(R_1\) link is temporarily added back into the normal tree. The corresponding equation is found using KVL (loop law).

The \(R_1\) link is removed, and the \(R_2\) link is temporarily added back into the normal tree. Once again, KVL is used to solve for the \(R_2\)’s secondary variable (\(V_{R 2}\)).

 
 

We have now written 4 elemental equations and 4 constraint equations. This set of 8 equations fully describes the system.

$$\begin{array}{c}{V_{R 1}=V_{S}-V_{C 1}} \\ {V_{R 2}=V_{C 1}-V_{C 2}} \\ {I_{C 1}=I_{R 1}-I_{R 2}} \\ {I_{C 2}=I_{R 2}}\end{array}$$

Other Notes

The equations developed here can be solved to find any property of the system, can be converted into block-diagram form to find the transfer function, or can be written in state-space form. This process was conducted for an electrical problem, however the process of developing elemental and constraint equations also applies to the mechanical (translational/rotational), thermal, and fluid domains.